Unimodular families of symmetric matrices

• Published in: arXiv.org
• Main Authors: Domitrz, Wojciech, Izumiya, Shyuichi, Teramoto, Hiroshi
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 31.05.2020
• Subjects: Canonical forms; Classification; Equivalence; Homogeneity; Isomorphism; Mathematical analysis; Matrix methods
• ISSN: 2331-8422

We introduce the volume-preserving equivalence among symmetric matrix-valued map-germs which is the unimodular version of Bruce's \(\mathcal{G}\)-equivalence. The key concept to deduce unimodular classification out of classification relative to \(\mathcal{G}\)-equivalence is symmetrical quasi-homogeneity, which is a generalization of the condition for a \(2 \times 2\) symmetric matrix-valued map-germ in Corollary~2.1 (ii) by Bruce, Goryunov and Zakalyukin. If a \(\mathcal{G}\)-equivalence class contains a symmetrically quasi-homogeneous representative, the class coincides with that relative to the volume-preserving equivalence (up to orientation reversing diffeomorphism in case if the ground field is real). By using that we show that all the simple classes relative to \(\mathcal{G}\)-equivalence in Bruce's list coincides with those relative to the volume preserving equivalence. Then, we classify map-germs from the plane to the set of \(2 \times 2\) and \(3 \times 3\) real symmetric matrices of corank at most \(1\) and of \(\mathcal{G}_e\)-codimension less than \(9\) and we show some of the normal forms split into two different unimodular singularities. We provide several examples to illustrate that non simplicity does not imply non symmetrical quasi-homogeneity and the condition that a map-germ is symmetrically quasi-homogeneous is stronger than one that each component of the map-germ is quasi-homogeneous. We also present an example of non symmetrically quasi-homogeneous normal form relative to \(\mathcal{G}\) and its corresponding formal unimodular normal form.